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We develop a maximum likelihood estimating approach for time-to-event Weibullregression models with outcome-dependent sampling, where sampling of subjectsis dependent on the residual fraction of the time left to developing the eventof interest. Additionally, we propose a two-stage approach which proceeds byiteratively estimating, through a pseudo score, the Weibull parameters ofinterest (i.e., the regression parameters) conditional on the inverseprobability of sampling weights; and then re-estimating these weights (giventhe updated Weibull parameter estimates) through the profiled full likelihood.With these two new methods, both the estimated sampling mechanism parametersand the Weibull parameters are consistently estimated under correctspecification of the conditional referral distribution. Standard errors for theregression parameters are obtained directly from inverting the observedinformation matrix in the full likelihood specification and by eithercalculating bootstrap or robust standard errors for the hybrid pseudoscore/profiled likelihood approach. Loss of efficiency with the latter approachis considered. Robustness of the proposed methods to misspecification of thereferral mechanism and the time-to-event distribution is also briefly examined.Further, we show how to extend our methods to the family of parametrictime-to-event distributions characterized by the generalized gammadistribution. The motivation for these two approaches came from data on time tocirrhosis from hepatitis C viral infection in patients referred to theEdinburgh liver clinic. We analyze these data here.

Maximum likelihood and pseudo score approaches for parametric time-to-event analysis with informative entry times